菜单
Explore a mathematical conjecture - gather evidence, find counterexamples, or build toward a proof. Triggers on "explore", "investigate", "what if", "is it true that", "find pattern", "check if"
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基于 SOC 职业分类
| name | conjecture-exploration |
| description | Explore a mathematical conjecture - gather evidence, find counterexamples, or build toward a proof. Triggers on "explore", "investigate", "what if", "is it true that", "find pattern", "check if" |
Systematically investigate a mathematical conjecture before attempting a proof.
symbolic_compute: Parse and simplify the statementsequence_lookup: Check if related to known sequencesweb_search: Check if this is a known resultFor integer conjectures, compute:
| n | LHS | RHS | Equal? |
|---|---|---|---|
| 0 | ... | ... | ✓/✗ |
| 1 | ... | ... | ✓/✗ |
| ... | ... | ... | ... |
symbolic_compute: Calculate values for test casesBased on evidence gathered:
If all cases pass + pattern is clear: → Suggest proof strategy (induction, direct, etc.) → Transition to appropriate proof skill
If counterexample found: → Report the counterexample with details → Ask: "Should we explore a weaker version?"
If pattern unclear: → Checkpoint with user → Share all findings → Ask for guidance
Present structured summary:
Determine convergence or divergence of series using ratio test, root test, comparison test, and integral test. Triggers on "series converges", "test for convergence", "ratio test", "root test", "comparison test", "integral test", "sum of 1/n"
Prove geometric theorems using coordinate geometry. Convert shapes to algebraic expressions, use distance formula, midpoint formula, and slope calculations. Triggers on "prove using coordinates", "analytic geometry", "distance formula", "show collinear", "prove perpendicular", "prove parallel"
Prove identities or existence using combinatorial counting - double counting, bijections, inclusion-exclusion. Triggers on "count", "number of ways", "how many", "bijection", "one-to-one correspondence", "combinatorial proof"
Prove statements directly by assuming the hypothesis and deriving the conclusion through logical steps. Triggers on "prove directly", "show that", "demonstrate", "verify", "simple proof"
Prove limit statements using epsilon-delta definitions. Triggers on "epsilon-delta", "limit", "prove continuous", "show limit equals", "formal limit proof"
Explain a proof or Lean code step-by-step for learning. Triggers on "explain", "how does this work", "teach me", "what does this mean", "walk me through", "I don't understand"